Chaos, concentration and multiple valleys in first-passage percolation

TL;DR

This paper explores the relationship between fluctuations, chaos, and multiple valleys in first-passage percolation, extending ideas from disordered systems to understand path sensitivity and phase transitions.

Contribution

It establishes a connection between fluctuations and chaos in first-passage percolation, identifying the stability-chaos transition and the emergence of multiple valleys.

Findings

Small perturbations cause significant changes in optimal paths.

The system exhibits a transition point from stability to chaos.

Chaotic behavior leads to many nearly optimal, disjoint paths.

Abstract

A decade and a half ago Chatterjee established the first rigorous connection between anomalous fluctuations and a chaotic behaviour of the ground state in certain Gaussian disordered systems. The purpose of this paper is to show that Chatterjee's work gives evidence of a more general principle, by establishing an analogous connection between fluctuations and chaos in the context of first-passage percolation. The notion of `chaos' here refers to the sensitivity of the time-minimising path between two points when exposed to a slight perturbation. More precisely, we resample a small proportion of the edge weights, and find that a vanishing fraction of the edges on the time-minimising path still belongs to the time-minimising path obtained after resampling. We also identify the point at which the system transitions from being stable to being chaotic in terms of the variance of the system. Finally we show that the chaotic behaviour implies the existence of a large number of almost-optimal paths that are almost disjoint from the time-minimising path, a phenomenon known as 'multiple valleys'.